Automata on ordinals and automaticity of linear orders

@article{Schlicht2013AutomataOO,
  title={Automata on ordinals and automaticity of linear orders},
  author={Philipp Schlicht and F. Stephan},
  journal={Ann. Pure Appl. Log.},
  year={2013},
  volume={164},
  pages={523-527}
}
Abstract We investigate structures recognizable by finite state automata with an input tape of length a limit ordinal. At limits, the set of states which appear unboundedly often before the limit are mapped to a limit state. We describe a method for proving non-automaticity and apply this to determine the optimal bounds for the ranks of linear orders recognized by such automata. 
The isomorphism problem for tree-automatic ordinals with addition
TLDR
An algorithm is described that, given two tree-automatic ordinals with the ordinal addition operation, decides if the ordinals are isomorphic. Expand
Structures without Scattered-Automatic Presentation
TLDR
This paper proves the following limitations on the class of \(\mathfrak{L}\)-automatic structures for a fixed \(\ mathfrak {L}\) of finite condensation rank 1 + α. Expand
L O ] 2 0 Ju l 2 01 7 Space-bounded OTMs and REG ∞
An important theorem in classical complexity theory is that LOGLOGSPACE=REG, i.e. that languages decidable with doublelogarithmic space bound are regular. We consider a transfinite analogue of thisExpand
The Field of the Reals and the Random Graph are not Finite-Word Ordinal-Automatic
TLDR
This work lifts Delhomm\'e's relative-growth-technique from the automatic and tree-automatic setting to the ordinal- automatic setting, which implies that the random graph is not Ordinal-automatic and infinite integral domains are not ordinals below $\omega_1+\omega^ \omega$ where $\omegas_1$ is the first uncountable ordinal. Expand
Tree-automatic scattered linear orders
TLDR
It is shown that there is no tree-automatic scattered linear order, and therefore no tree's automatic well-order, on the set of all finite labeled trees, and that a regular tree language admits a tree- automatic scattered linear orders if and only if for some n, no binary tree of height n can be embedded into the union of the domains of its trees. Expand
Pumping for ordinal-automatic structures
TLDR
A pumping lemma for alpha-automata (processing finite alpha-words, i.e., words of length alpha that have one fixed letter at all but finitely many positions) is developed and a sharp bound on the height of the finite word alpha-automatic well-founded order forests is provided. Expand
Space-Bounded OTMs and REG$^{\infty}$
An important theorem in classical complexity theory is that LOGLOGSPACE=REG, i.e. that languages decidable with double-logarithmic space bound are regular. We consider a transfinite analogue of thisExpand

References

SHOWING 1-10 OF 21 REFERENCES
Automata on Ordinals and Linear Orders
TLDR
This work investigates structures recognizable by α-automata with running time a limit ordinal α and determines the suprema of the α-automatic ordinals and the ranks of α- automatic linear orders. Expand
Automata on linear orderings
TLDR
It is proved that for countable scattered linear orderings, the two notions of finite automata and rational expressions are equivalent, which extends Kleene's theorem. Expand
Accessibility in Automata on Scattered Linear Orderings
TLDR
If only words indexed by scattered linear orderings are considered, the accessibility and the emptiness in these automata can be checked in time nm 2, which solves the problem for automata on transfinite words. Expand
Automatic linear orders and trees
TLDR
It is shown that every infinite path in an automatic tree with countably many infinite paths is a regular language. Expand
Finite Automata, Definable Sets, and Regular Expressions over omega^n-Tapes
  • Y. Choueka
  • Computer Science, Mathematics
  • J. Comput. Syst. Sci.
  • 1978
TLDR
The theory of finite automata and regular expressions over a finite alphabet Σ is generalized to infinite tapes X = X 1 … X k, where X i, are themselves tapes of length ω n, and the equivalence of deterministic and nondeterministic automata is proved. Expand
Logic Colloquium 2007: Three lectures on automatic structures
This paper grew out of three tutorial lectures on automatic structures given by the first author at the Logic Colloquium 2007. We discuss variants of automatic structures related to several models ofExpand
Interpretations in Trees with Countably Many Branches
  • A. Rabinovich, S. Rubin
  • Mathematics, Computer Science
  • 2012 27th Annual IEEE Symposium on Logic in Computer Science
  • 2012
TLDR
It is proved, roughly, that trees and orders of large rank cannot be interpreted in scattered trees of small rank, and how to replace injective set-interpretations in (not necessarily scattered) trees by âfinitary' set- interpretations is shown. Expand
On a Decision Method in Restricted Second Order Arithmetic
Let SC be the interpreted formalism which makes use of individual variables t, x, y, z,... ranging over natural numbers, monadic predicate variables q( ), r( ), s( ), i( ),... ranging over arbitraryExpand
Tree-Automatic Well-Founded Trees
TLDR
It is shown that the isomorphism problem for tree-automatic well-founded trees is complete for level $\Delta^0_{\omega^ \omega}$ of the hyperarithmetical hierarchy (under Turing-reductions). Expand
Automaticité des ordinaux et des graphes homogènes
Resume Les structures automatiques (resp. arbre-automatiques) sont les structures relationnelles dont le domaine est un ensemble regulier de mots (resp. de termes) finis et dont chaque relationExpand
...
1
2
3
...